Directional Pliability, Whitney Extension, and Lusin Approximation for Curves in Carnot Groups

TL;DR

This paper establishes that in Carnot groups, directional pliability ensures Whitney extensions and Lusin approximations for curves, with applications to intersections of horizontal curves in the Engel group.

Contribution

It introduces the concept of pliability in Carnot groups and proves its sufficiency for Whitney and Lusin approximation theorems for curves.

Findings

Pliability in directions guarantees Whitney extension in Carnot groups.

Pliability ensures Lusin approximation for curves with tangent vectors in the same directions.

Every horizontal curve in the Engel group intersects a $C^{1}$ horizontal curve in a set of positive measure.

Abstract

We show that, in arbitrary Carnot groups, pliability in a subset of directions is sufficient to guarantee the existence of a Whitney-type extension and a Lusin approximation for curves with tangent vectors in the same set of directions. We apply this to show that every horizontal curve in the Engel group must intersect a $C^{1}$ horizontal curve in a set of positive measure.